Lemma 26.7.5 (01IB)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 26: Schemes
Section 26.7: Quasi-coherent sheaves on affines
Lemma 26.7.5 (cite)

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Lemma 26.7.5. Let $(X, \mathcal{O}_ X) = (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ be an affine scheme. The functors $M \mapsto \widetilde M$ and $\mathcal{F} \mapsto \Gamma (X, \mathcal{F})$ define quasi-inverse equivalences of categories

\[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_ X) \ar@<1ex>[r] & \text{Mod}_ R \ar@<1ex>[l] } \]

between the category of quasi-coherent $\mathcal{O}_ X$-modules and the category of $R$-modules.

Proof. See Lemmas 26.7.1 and 26.7.4 above. $\square$

Comments (2)

Comment #427 by Jeroen Zuiddam on January 13, 2014 at 20:37

I am getting a parse error in the display.

Comment #428 by Johan on January 13, 2014 at 22:12

OK, I think this is due to an XyJax parsing error. I'll investigate more. Thanks for the report!

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